1. Improved Small-x Evolution with Running Coupling Effects DIS03, 24 April ‘03 CERN G. Altarelli Based on G.A., R. Ball, S.Forte hep-ph/9911273 (NPB 575,313) hep-ph/0001157 (lectures) hep-ph/0011270 (NPB 599,383) hep-ph/0104246 More specifically on hep-ph/0109178 (NPB 621,359) and on CERN-TH/2003-019 now being completed

2. G. Altarelli Our goal is to construct a relatively simple, closed form, improved anom. dim.  I ( ,N)  I ( ,N) should reduce to pert. results at large x contain BFKL corr’s at small x include running coupling effects be sufficiently simple to be included in fitting codes closely follow the trend of the data and of course

3. G. Altarelli Moments For each moment: singlet eig.vector with largest anomalous dim. eig.value Singlet quark Mellin transf. (MT) Inverse MT (  >0) t-evolution eq.n  : anom. dim known Pert. Th.: LONLO

4. G. Altarelli In principle the BFKL approach provides a tool to control (  /N) n corrections to  (N,  ), that is 1/x(  log1/x) n to splitting functions. Define t- Mellin transf.: With inverse:  -evolution eq.n (BFKL): with known Bad behaviour Bad convergence

5. G. Altarelli  M  s =0.2  0 +  2  1  0 The minimum value of  0 at M=1/2 is the Lipatov intercept: 0 =  0 (1/2)=  c 0 =  4n c /  log2~2.65  ~0.5 It corresponds to (for x->0): xP(x)~x - 0 Too hard not supported by data

6. G. Altarelli In the region of t and x where both are approximately valid, the "duality" relation holds: Note:  is leading twist while  is all twist. Still the two perturbative exp.ns are related and can improve each other. Non perturbative terms in  correspond to power or exp. suppressed terms in 

7. G. Altarelli For example at 1-loop:  0 (   ( ,N))=N/   0 improves  by adding a series of terms in  /N) n   0 -> Similarly it is very important to improve  by using  1l. Near M=0  0 ~ 1/M,  1 ~ -1/M 2 Duality + momentum cons. (  ( ,N=1)=0) { Double Leading Expansion

8. G. Altarelli  The DL expansion is OK near M=0. But is still bad near M=1 (no analogue of mom. cons.). One can try to use symmetry of underlying BFKL kernel (broken by scale unbalance in DIS) Ciafaloni,, Salam +Colferai, Camici or introduce some additional idea to treat the central region GA, Ball, Forte

9. G. Altarelli The implementation of running coupling in BFKL is not simple. In M-space  -> operator In leading approximation: A perturbative expansion in  0 leads to validity of duality with modified  and  But expansion fails near M=1/2:  0 '(1/2)=0

10. G. Altarelli DL-LO DL-NLO Res At M=1/2  0 has a minimum and  1 is singular (and also  ss ). We shall see it is just an artifact of pert. exp.

11. G. Altarelli By taking a second MT the equation can be written as [F(M) is a boundary condition] It can be solved iteratively or in closed form: H(N,M) is a homogeneous eq. sol. that vanishes faster than all pert. terms and can be dropped.

12. G. Altarelli The following properties can be proven: From G(N,M) we can obtain G(N,t) and evaluate it by saddle point expansion. This G(N,t) satisfies duality (in terms of modified  and  according to the perturbative results singular at  '(1/2)=0) and factorisation(no t-dep. from the boundary condition). From G(N,M) we can obtain G( ,M). This presents unphysical oscillations when  >0 for all M. These problems can be studied by using the Airy approximation: The asymptotics is fixed by the behaviour of  near the minimum, where a quadratic form is taken (here c is free and k as in    : Lipatov; Collins,Kwiecinski;Thorne; Ciafaloni, Taiuti,Mueller G.A., R. Ball, S.Forte hep-ph/0109178 (NPB 621,359)

13. G. Altarelli The explicit solution is where Ai''(z)-zAi(z) = 0 Ai(0) = 3 -2/3  (2/3) Ai(z) z z 0 ~-2.338

14. G. Altarelli From one obtains G(x,t) by inv. MT The asymptotics is dominated by the saddle condition: For c>0 at not too large  this is satisfied at large N. When  increases N gets smaller. Then oscillations start, d/dN changes sign and the real saddle is lost. G( ,t) starts oscillating, in agreement with the general analysis.

15. G. Altarelli Ai[z(N)] N c= -1  =0.2 N Ai[z(N)] c= +1  =0.2

16. G. Altarelli The dual anom. dim.  A is given by N large N0N0 z0z0 +... c=1  s =0.2

17. G. Altarelli The splitting function is completely free of oscillations at all x!! 2-loop c=+2 +1 0 -2 The oscillations get factorised into the initial condition

18. G. Altarelli The Airy result is free of the perturbative  0 singularities. At NLL order we can add the full  A and subtract its large N limit: The last term cancels the sing. of  ss (N=  c corresponds to M=1/2)  0 ->  s,  1 ->  ss

19. G. Altarelli The effect of running on  in the Airy model Is a softer small-x behaviour c=1 k=-2n c /  ’’ (1/2) =32.14.. cc N0N0 xP ~ x - xP ~ x - N 0

20. G. Altarelli c=2, 1.5, 1, 0.5 =c  s (Q 2 ) N 0 (Q 2 ) As an effect of running, the small-x asymptotics is much softened: xP ~ x - xP ~ x - N 0

21. G. Altarelli The goal of our ongoing work is to use these results to construct a relatively simple, closed form, improved anom. dim.  I ( ,N)  I ( ,N) should reduce to pert. result at large x contain BFKL corr’s at small x include running coupling effects (Airy) be sufficiently simple to be included in fitting codes closely follow the trend of the data and of course here I will describe the LO approximation G.A., R. Ball, S.Forte, CERN-TH/2003- 019

22. G. Altarelli Improved anomalous dimension 1st iteration: optimal use of  1l (N) and  0 (M) Properties: Pert. Limit  ->o, N fixed o(  2 ) Limit  ->o,  /N fixed +o(   ) Pole in 1/N Cut with branch in  c 0 the Airy addition cancels the cut and replaces it with a pole at N=N 0

23. G. Altarelli  II GLAP LO&NLO  DL  s =0.2  DL ( , N )=

24. G. Altarelli xP 1/x GLAP LO&NLO II  DL  s =0.2 Here we have the same plot for the corresponding splitting functions. Note: for  s =0.2 the pole in GLAP is ~0.191/N while the pole in  I is ~0.014/(N-N 0 ) (only visible at very small x)

25. G. Altarelli Conclusion BFKL with running coupling is fully compatible with RGE t evolution, factorisation and duality. Using the Airy solution we have seen that the splitting functions are completely free of unphysical oscillations (can be factorised in the initial condition at t 0 ). The Airy solution can be used to resum the perturbative singularities in the  0 expansion We use these results to construct an improved an. dim. that reduces to the pert. result at large x and incorporates BFKL with running coupling effects at small x. Properly introducing running coupling effects in the LO softens the asymptotic small x behaviour as indicated by the data.